Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 4 / 7F / d / Solution

DefineThen $\mathbf{x}=\mathbf{y}+\mathbf{z}$, the support of $\mathbf{y}$ lies in $P(\mathbf{s})$, and the support of $\mathbf{z}$ is disjoint from $P(\mathbf{s})$. Thus $\mathbf{y}⪯\mathbf{s}$ and $\mathbf{z}\perp \mathbf{s}$.

For uniqueness, if $s_i>0$, orthogonality forces $z_i=0$, hence $y_i=x_i$. If $s_i=0$, the condition $\mathbf{y}⪯\mathbf{s}$ forces $y_i=0$, hence $z_i=x_i$. Every coordinate is therefore forced, proving uniqueness.